Question

In a region of two-dimensional space, there are three fixed charges. \(+1.00 \mathrm{mC}\) at \((0,0),-2.00 \mathrm{mC}\) at \((17.0 \mathrm{~mm},-5.00 \mathrm{~mm}),\) and \(+3.00 \mathrm{mC}\) at \((-2.00 \mathrm{~mm}, 11.0 \mathrm{~mm}) .\) What is the net force on the \(-2.00-\mathrm{mC}\) charge?

Short answer

1. Identify all charged particles' magnitudes and positions and the unknown net force. 2. Calculate the electric force between each pair of charges using Coulomb's Law. 3. Calculate the net force on the given charge by obtaining the vector sum of the relevant forces.

Step-by-step solution

Identify all required parameters and unknowns

In our case, our known values are: - Charge 1: \(q_1 = +1.00 \ \text{mC}\) at position \(A(0, 0)\) - Charge 2: \(q_2 = -2.00 \ \text{mC}\) at position \(B(17.0 \ \text{mm}, -5.00 \ \text{mm})\) - Charge 3: \(q_3 = +3.00 \ \text{mC}\) at position \(C(-2.00 \ \text{mm}, 11.0 \ \text{mm})\) The unknown is the net force on Charge 2.

Calculate the electric force between each pair of charges

Using Coulomb's Law, the electric force between two charges \(q_1\) and \(q_2\) is given by: \(F = k\frac{|q_1q_2|}{r^2}\), where \(k\) is the Coulomb's constant \(=\ 8.99\times10^9 \ \text{N}\cdot\text{m}^2/\text{C}^2\), \(r\) is the distance between the charges. Calculate the forces and their directions as follows: 1. Force between Charge 1 and Charge 2: \(F_{12} = k\frac{|q_1q_2|}{r_{12}^2}\), where \(r_{12} = |A - B|\). 2. Force between Charge 2 and Charge 3: \(F_{23} = k\frac{|q_2q_3|}{r_{23}^2}\), where \(r_{23} = |B - C|\). 3. Force between Charge 1 and Charge 3: \(F_{13} = k\frac{|q_1q_3|}{r_{13}^2}\), where \(r_{13} = |A - C|\). Note that while we calculate \(F_{13}\), it will not be needed to find our net force on Charge 2.

Calculate the net force on Charge 2

To find the net force on Charge 2, we need to get the vector sum of \(F_{12}\) and \(F_{23}\). 1. Convert the magnitudes \(F_{12}\) and \(F_{23}\) into their respective vector components, \(F_{12x}, F_{12y}, F_{23x},\) and \(F_{23y}\). 2. Add the corresponding components: \(F_{netx} = F_{12x} + F_{23x}\), and \(F_{nety} = F_{12y} + F_{23y}\). 3. Find the magnitude of the net force: \(F_{net} = \sqrt{F_{netx}^2 + F_{nety}^2}\). This will give us the net force on the \(-2.00 \ \text{mC}\) charge.